Harlene tosses two number cubes. If a sum of 8 or 12 comes up, she gets 9 points. If not, she loses 2 points. What is the expected value of the number of points for one roll?

A. [tex]\(-\frac{2}{3}\)[/tex]
B. [tex]\(-\frac{1}{6}\)[/tex]
C. [tex]\(\frac{1}{6}\)[/tex]
D. [tex]\(\frac{2}{3}\)[/tex]



Answer :

To determine the expected value of the number of points Harlene will receive for one roll, we'll break down the question into several steps:

1. Identify all possible outcomes and desired outcomes:
- Each number cube has 6 faces, resulting in 36 (6 * 6) possible outcomes when two dice are rolled.

2. Calculate the number of ways to get sums of 8 and 12:
- Sum of 8: The possible combinations to get this sum are: (2,6), (3,5), (4,4), (5,3), and (6,2). This gives us 5 combinations.
- Sum of 12: The only way to get this sum is (6,6), which accounts for 1 combination.
- Therefore, there are a total of 5 (sum of 8) + 1 (sum of 12) = 6 ways to get either a sum of 8 or 12.

3. Determine the probabilities:
- Probability of winning (sum of 8 or 12):
[tex]\[ \frac{6}{36} = \frac{1}{6} \approx 0.16666666666666666 \][/tex]
- Probability of losing (any other sum):
Since there are 36 possible outcomes and only 6 of those are wins, the rest are losses:
[tex]\[ 1 - \frac{1}{6} = \frac{5}{6} \approx 0.8333333333333334 \][/tex]

4. Calculate the points:
- Points for winning (getting a sum of 8 or 12): +9 points
- Points for losing (any other outcome): -2 points

5. Compute the expected value:
To calculate the expected value (E), we use the formula:
[tex]\[ E = (\text{Points for Win} \times \text{Probability of Win}) + (\text{Points for Loss} \times \text{Probability of Loss}) \][/tex]

Plugging in the numbers from above:
[tex]\[ E = (9 \times \frac{1}{6}) + (-2 \times \frac{5}{6}) \][/tex]
Simplify each term:
[tex]\[ E = 1.5 - 1.6666666666666667 \approx -0.16666666666666674 \][/tex]

The final expected value of the points for one roll is approximately [tex]\(-0.16666666666666674\)[/tex], which corresponds to the decimal form of [tex]\(-\frac{1}{6}\)[/tex].

Hence, the correct choice is:
[tex]\[ -\frac{1}{6} \][/tex]